Vrton o AG Struturs n Cooprtv B Ntwor Bs Mut-nt Systs Y. Xn prtnt o Coputr Sn, Unvrsty o Rn Rn, Sstwn, Cn S4S 0A, yxn@s.urn. Astrt Mutpy ston Bysn ntwors (MSBNs) prov rwor or prost rsonn n opx sn usr ornt syst s w s n ooprtv ut-nt strut ntrprtton syst. urn t onstruton or yn orton o MSBN, utot vrton o t yty o t ovr strutur s sr. W nown orts or yty tst ssu ntrz stor o t strutur to tst. W suss wy ntrz tst s unsr n propos strut ort tt vrs t yty trou ooprton on sunts/nts. T ort os not rqur nt to rv ts ntrn strutur n tus supports onstruton o MSBN ro sunts ut y rnt vnors. Introuton Mutpy ston Bysn ntwors (MSBNs) s n xtnson o Bysn ntwors (BNs) [, 4, ]. A MSBN onssts o st o ntrrt Bysn sunts tt otvy n BN [, ]. E sunt srs non-pty st o vrs wt t st on otr sunt. Sunts r ornz nto yprtr strutur su tt prost nrn n pror ornty n our n strut son. T ourty provs nrn ny n opx sn usr ornt syst [0]. It so ows MSBNs to xtn nto ornt rwor or prost rsonn n ooprtv ut-nt strut ntrprtton systs [8]. T strutur o BN s rt y rp (AG). T ovr strutur o MSBN, t oposton o sunt struturs, s so AG. To nsur t orrt oposton, utot vrton o yty o t opos strutur s sr. Atou orts or tstn yty s on topoo sortn r w nown, s [6] or xp, ty ssu ntrz stor o t rp to tst. W nyz so sn onsrtons tt ntrz tst unsr. W tn propos strut ort n trs o st o strut oprtons or tstn yty o t opos strutur trou ooprton o sunts/nts. T tory n pptons o MSBNs r ry rvw n Ston. T onpts nssry to t rst o t ppr r ory n. W suss n Ston rsons
wy strut vrton o yty s prrr. It s sown n Ston 4 tt so ovous soutons to strut vrton o not sov t pro. T rp-tort ounton or t propos ort s rv n Ston 5 n t ort s prsnt n Ston 6 wt proo o ts orrtnss. Its opxty s nyz n Ston 7. Ovrvw o MSBNs In ts ston, w ry ovrvw t tory o MSBNs n tr pptons. Mor ts on MSBNs n oun n [, 0, 7, 8]. ABNS s trpt (N,,P) wr N s st o vrs, s AG wos nos r y nts o N, n P s ont proty struton (p) ovr N. W s N t on o S, t strutur o S n P t struton or p o S. A MSBN M s oton o Bysn sunts tt totr n BN. Ts sunts r rqur to stsy rtn ontons tt prt t onstruton o strut nrn orts. On o ts ontons rqurs tt nos sr y rnt sunts or -spst, s n ow. Lt G =(N,E )( =, ) two rps. W s rr to t rp G =(N N,E E ) s t unon o G n G, not y G = G G. nton (-spst) Lt =(N,E )( =, ) two AGs su tt = s AG. T ntrston I = N N s -spst twn n or vry A I wt ts prnts π n, tr π N or π N. E no n -spst s -spno. T -spst onpt s syntt onton. Snty, t n sown tt wn pr o sunts r sot ro M, tr -spst rnrs t ontony npnnt. Tror, -spst provs sp syntt ru to tt npnnt spton o snty orrt sunts. Fur (t) sows t tr AGs ( =,, ) o MSBN or noss o tr nurousur sss, Mn nrv son (Mn), Crp tunn synro (Cts) n Pxus uppr trun son (Pxut). T -spst twn pr o AGs s {Mn, Cts, P xut}. In nr, -spsts twn rnt prs o AGs o M y rnt. Just s t strutur o BN s AG, t strutur o MSBN s utpy ston AG (MSAG) wt yprtr ornzton, or spy yprtr MSAG n s oows: nton (Hyprtr MSAG) A yprtr MSAG =, wr s onnt AG, s AG tt s ut y t oown prour: Strt wt n pty rp (no no). Rursvy AG,yprno, to t xstn MSAG = sut to t onstrnts: [-spst] For (<), t ntrston I = N N s -spst wn t two AGs r sot. T xp s tn ro rton o PAINULIM [0] o or ustrton.
Pxut pns Mn wr oup Cts pn Pxut ps Mn prt Cts p Pxut Mn Cts p up {Mn,Cts, Pxut} {Mn,Cts,Pxut} 6 4 5 7 Fur : Lt: T AGs o n xp MSBN, wr -spno s t y ott r. M: T yprtr ornzton o t AGs n t t. Rt: A nr yprtr MSAG (unrt to t t). [Lo ovrn] Tr xsts (<) su tt, or (<; ), w v I N. For n rtrry osn su, I s t yprn twn yprnos n, n n r s to nt. Not tt yprtr MSAG s tr wr no s yprno s n ov n n s yprn. T AGs n Fur (t) n ornz nto t trv yprtr MSAG n Fur (), wr yprno s y AG n yprn s y -spst. Fur (rt) pts nr yprtr MSAG. Atou AGs o MSBN sou ornz nto yprtr, AG y utpy onnt (or tn on pt xst twn pr o nos),..,. Morovr, tr n utp pts twn pr o nos n rnt AGs n yprtr MSAG. For nstn, utp pts r or twn p n p tr n r unon. T o ovrn onton nsurs tt or ny unrt y ross two nt AGs, ot o ts two pts r trou t orrsponn -spst. Totr wt t -spst onton, ty nsur tt n yprtr strutur M, yprn rnrs t two prts o M tt t onnts ontony npnnt. An ntutv ustton o ts strutur s vn n [9]. A MSBN s n s oows. Rrs r rrr to [] or or ts. nton A MSBN M s trpt (N,, P). N = N s t tot unvrs wr N s st o vrs. = ( yprtr MSAG) s t strutur wr nos o AG r y nts o N. P = P (N )/ P (I ) s t ont proty struton (p). E P (N ) s proty struton ovr N su tt wnvr n r nt n, t rnztons o P (N ) n P (N ) onto t -spst I r nt. E P (I ) s su rn struton ovr yprn o. E trpt S =(N,,P ) s sunt o M. Wtout onuson, w s sy tt two sunts S n S r nt n r nt. A MSBN n us s rwor or prost rsonn n sn usr ornt syst n r pro on. T sn usr ps tt vn n qurs r rstrt to on suon t t. Usn MSBN s ost n suons o t pro on r oosy oup (t sz o -spst s rsony s rtv
to t sz o t suon) n vn n qurs r ous on on suon or pro o t or stn to rnt suon. For xp n Fur (rt), t usr y ous ttnton on sunt S wos strutur s. Atr svr ps o vn r ntr n qurs r ssu to ts sunt, t usr y st ttnton to t sunt S. T nrn oprtons o MSBNs w tn propt vn ro S to S n tn to S. T usr n tn ntr vn on vrs ontn n S. It n sown tt wt su rstrt propton urn ttnton st, t nswrs to qurs otn n S r wys onsstnt to vn uut n t ntr MSBN. Coputton opxty, owvr, s ru y not vn to upt ny sunts not on t yprpt ro t urrnt sunt to t nxt trt sunt. Appton ons o sn-usr MSBNs nu noss o ntur systs [0] n o-s noss o rt systs [5]. Snsors Bysn Sunt son Mr Atutor Snstvty Anyzr Rsonr Countor strut Go Strutur Vrr Fur : Mn oponnts o n nt n MSBN-s ut-nt syst. MSBNs n xtn nto rwor or prost rsonn n ooprtv ut-nt strut ntrprtton systs. E nt os ts prt prsptv o r pro on (Sunt n Fur ), sss o vn sour (Snsors n Fur ), ounts wt otr nts nrqunty (Countor), rsons wt t o vn n t o vn (Rsonr), n nswrs qurs (Rsonr) or ts tons (son Mr/Atutor). It n sown [8] tt nts r ooprtv (vs s-ntrst), n pr o nt nts r ontony npnnt vn tr sr vrs n v oon nt on t sr vrs, tn ont syst s w n w s nt to nt s wtn ts suon n suppnt to t nt s outs t suon. Evn tou utp nts y qur vn synronousy n pr (opr wt t sn usr ornt syst wr vn s wys ntr nto t urrnt sunt o ous), t ounton oprtons o MSBNs nsur tt t nswrs to qurs ro nt r onsstnt wt vn qur n t ntr syst tr ounton. Sn ounton s nrqunt, t oprtons so nsur tt twn two sussv ountons, t nswrs to qurs or nt r onsstnt wt o vn tr so r n r onsstnt wt vn tr n t ntr syst up to t st ounton. Tror, MSBN n rtrz s on o untony urt, ooprtv strut systs []. Potnt pptons nu son support to ooprtv un Atou MSBNs r not rrn rty, t rprsntton ors us s sp s o MSBNs. For xp, t st o nput nos I, output no O, o no M, n uy no [5], w ors n ntr twn r v n owr v n t rry, s -spst []. T opost ont tr [5] orrspons to t yprtr []. T wy n w nrn s pror n t opost on tr orrspons to t oprton StAttnton []. 4
usrs n unrtn ons n trousootn opx syst y utp now s susysts [8]. Wy strut Vrton? As n n Ston, t strutur o MSBN s yprtr MSAG w sou AG. Autot vrton o yty o ts strutur s sr n t onstruton o r MSBNs. Aorts tt tst wtr rt rp s AG s on topoo sortn r w nown, s or xp [6]. Ts orts, owvr, ssu ntr rprsntton o t rp strutur to tst. A ntr rprsntton o AGs n MSBN s not sr or t st two rsons. Frst, t onstruton o ut-nt MSBN rqurs ony t now o t untonty o sunt n t ntr (-spst) twn sunts (BNs). Known t ntrn strutur o sunt s not nssry. Tror, sunt y vop y n npnnt vnor wo y not wn to sos t strutur ts. T ssupton o ntr rprsntton o AGs w nt t possty o ooprtn nts ut y su vnors. Sony, MSBN n potnty yn. Tt s, sunts y on or v t MSBN s t syst s untonn. It s sr to vry t orrtnss o t strutur o t syst wnvr t r sunts n. It s so sr tt t vrton os not rqur t ounton o AGs to ntr oton or os not pn upon sn nt to ntn rpostory o AGs n t urrnt syst. In ts ppr, w propos strut ort or vrton o t yty o MSBN strutur. urn t vrton pross, nt ony provs nswrs to nt sunts on qustons rrn -spnos, n t os not rv ts ntrn strutur yon tt. 4 Issus n strut Vrton R ro nton tt yprtr MSAG s ut ro st o AGs sut to t -spst n o ovrn ontons. Howvr rt rp ut ro st o AGs sut to ts two ontons y st ontn rt ys. W s rr to t rsutnt rp s yprtr AG unon sn t y not quy s yprtr MSAG. Fur : A y AG unon. 5
Fur sows two AGs n wt tr -spst n {, }. I w unon t two AGs, t ry stss t o ovrn onton. Howvr, t unon ontns t rt y (,,,,, ) n tus s not AG. T ov y n tt w unon t pr o AGs n tst t yty. Atou t prws vrton y tt so rt ys, prws yty n yprtr AG unon os not urnt t o yty. n o Fur 4: Tr AGs w r prws y ut wos unon s y. Consr t tr AGs n Fur 4. T unon o n s y n so s t unon o n. Howvr, wn t tr AGs r unon, rt y {,,,, n,,,,, } s or. Cry, strut vrton o yty rqurs ooprton yon prs. 5 Vrton y Mrn Nos In ts ston, w sow tt yty o rt rp n vr y rn root n nos rursvy. On t s sts, w n r non--spnos oy n r -spnos y ooprton s prsnt n t nxt ston. A no s root r onnt to t s rt wy ro t. A no s r onnt to t s rt towr t. A no x s r x n rs onnt to x r nor ro urtr vrton pross. T oown two propostons sow tt rn o root/ nos os not n yty. Proposton 4 Lt G rt rp n x tr root or n G. Tn t yty o G rns tr x s r. Proo: I G s y, tn rn x nnot rt rt y n G. Suppos G s y. Tn tr xsts non-pty st O o rt ys n G. Ix s root, t os not v ny non r. I x s, t os not v ny outon r. Tror, x nnot prtpt n ny ys n O, w ps tt non o t ys n O w n tr x s r. On root or s r, otr nos y o roots or vs. Hn rn roots n vs n pror rursvy w prsrvn t yty. 6
Nxt, w sow tt rt rp s y, vry no n t w r y rursv pptons o Proposton 4. On t otr n, t s y, t st tr nos w t unr. Proposton 5 Lt G rt rp. G s y t s pty tr rursv rn o roots n vs. Proo: Wtout osn nrty, w ssu tt G s t st two nos n s onnt. Suppos G s y. Tn G s t st on root n on. Aorn to Proposton 4, tr o t r r, t rsutnt rp s st y n s nw roots n vs. Sn G s nt nur o nos, tr rursv rn o roots n vs, vntuy G w v no unr nos. Nxt, suppos G s y. Tn G s t st on rt y θ onsstn o t st tr nos. For no x n θ, t s ntr root nor n tus nnot r s su. Mrn o ny nos outs θ nnot turn x nto root or. Hn non o t nos n θ n r y rursv rn o roots n vs. Sn G s nt nur o nos, tr rursv rn o roots n vs, vntuy tr w no roots or vs to r n G w nos (t st tr) n θ r unr. W now onsr yprtr AG unon G wos nos r ss nto -spnos n non--spnos. Sn G s onnt rt rp, Proposton 5 n pp to tst ts yty. Howvr, tou non--spno roots n vs n ronz oy wtn sunt, -spno roots n vs n ony ronz trou ooprton on sunts. For xp, t no ( non--spno) n Fur 4 s ot n (pprn r oy) n n t AG unon (n r oy). On t otr n, ( -spno) s n (pprn r oy), root n (pprn r oy), ut non-root/non- n t AG unon (n not r oy). Morovr, rn o -spno roots n vs y turn so non--spnos nto nw roots or vs. T oown proposton sows tt rursv n trnt rn o non--spno roots/vs n -spno roots/vs s sunt to tst t yty o yprtr AG unon. Corory 6 Lt G yprtr AG unon. Lt G t rp rsutn ro rursv n trnt rn o non--spno roots/vs n -spno roots/vs n G unt no or nos n r. Tn G s y G s pty. Proo: Aorn to Proposton 5, G s y, t roun o rursv rn, tr so non--spno roots/vs or so -spno roots/vs n r, unt G s pty. I G s y, t roun o rursv rn, tr so non--spno roots/vs or so -spno roots/vs n r, unt ony nos n rt ys n G r t unr (t st tr). T trnt rn o -spnos n non--spnos s nssry. Otrws, t rn y t prtury vn G s y. To ustrt t nssty o trnt rn, onsr t y AG unon n Fur 5. Wtout usn trnt rn, w v ony two optons: () rursv rn o 7
non--spno roots/vs oow y rursv rn o -spno roots/vs, or () rursv rn o -spno roots/vs oow y rursv rn o non--spno roots/vs. Usn opton (), non--spnos (root) n () n w r n t rst st. Ntr o t non--spnos n n n r t ts st. In t son st, t -spnos (now root) n (now ) n r. T rn trnts wt n unr. Usn opton (), no -spno n r n t rst st sn ntr nor s root or. In t son st, non--spnos (root) n () n w r. T rn trnts wt,, n unr. Usn trnt rn y strtn wt non--spnos, n w r n t rst st. In t son st, -spnos n w r. In t tr st, non--spnos n w r. Now t rsutnt rp s pty. Atrnt rn y strtn wt -spnos vs t s rsut. Fur 5: A y AG unon. Not tt t orory os vn wn G s n rtrry AG unon (not sut to t two ontons n nton ), n -spnos r rp y sr nos. Su nrty s not n or our purpos. Corory 6 ors t ss or strut vrton ort w w prsnt n Ston 6. 6 Cooprtv Vrton As onstrt n Stons 4 n 5, n orr to vry t yty o yprtr AG unon, nts ust ooprt. Sn ooprton rqurs ounton w nurs ovr, t s sr to spy t ts or ooprton s u s poss. Aorn to Corory 6, non--spno roots/vs n yprtr AG unon n r sprty n rursvy. W n prprossn oprton to r ts nos or ooprton strts. Lt AG n yprtr AG unon G rtrry osn. I w trt ts AG s t root o t yprtr n rt t yprns o t yprtr wy ro t, tn t yprtr s onvrt nto rt tr. For vn AG, w n tn rr to nt AG s ts or ts prnt n t nor sns. Oprton 7 (PrPross) Wn PrPross s n AG, t oown r pror: 8
. rursvy rs non--spno root or.. s PrPross n AG. Atr PrPross s opt n G, nos t unr n AG r tr sot -spnos, or nos tt or rt pts n wt -spnos. Cooprton on AGs s n to urtr t vrton pross. Fur 6 sows t tr AGs n Fur 4 tr PrPross s ntt n ny o t. Ony rt pts r t n ts s. W w s sot -spnos n tr xp. n o Fur 6: T AG unon n Fur 4 tr o prprossn. Mr nos r sown s ry. To n out -spno x n r, AG uss t oprton CotFyIno to trn x s root or trou ooprton. T oprton psss trp (x, p, ) roun AGs w ontn x. T purpos s to ot t prnt/ norton or x, wr p s ount o t nur o AGs tt ontn prnts o x n s ount o t nur o AGs tt ontn rn o x. T r n t oown nton rrs to tr prnt AG or t nxt r v o oprton w ntt ts oprton. Oprton 8 (CotFyIno) Wn CotFyIno(x) s n AG, t oown r pror:. ors trp t 0 =(x, p 0, 0 ), wr p 0 = ontns n (unr) prnt o x n p 0 =0otrws, n 0 = ontns n (unr) o x n 0 =0 otrws.. I s no AG to w x s -spno, or p 0 = 0 =, tn rturns t 0 to r.. Otrws, s CotFyIno(x) n AG to w x s - spno. 4. Atr AG n s rturn tr trps (ssun AGs r ), t,t,...,t, rturns trp t =(x, p = x =0 p,= x =0 ) to r. On -spno x s trn to root (p = 0) or ( = 0), t oprton strutmr s us to r t n vry AG tt ontns t. 9
Oprton 9 (strutmr) Wn strutmr(x) s n AG, t oown r pror:. rs t no x.. rursvy rs ny non--spno root or.. I s ny nt AG to w x s -spno xpt r, tn s strutmr(x) n o t. Not tt n ts oprton t rn o -spno s trnt wt t rn o non--spnos s rqur y Corory 6. T oprton MrNo ons CotFyIno n strutmr to pror on roun o rn o -spnos. It rs -spno root/ own t yprtr, n trnts rn wt t rn o non--spno roots/vs n AG. Oprton 0 (MrNo) Wn MrNo s n AG, t oown r pror:. rturns s t s no AG, otrws ontnus.. For unr -spno x wt AG o, s CotFyIno(x) n ts. Wn t trp (x, p, ) s rturn to, s strutmr(x) n ts p =0or =0.. s MrNo n AG. 4. I ny AG rturns tru or strutmr(x) ws n, tn rturns tru to r. Otrws, rturns s (no no s r). T oprton MrA s wtr nos n yprtr AG unon v n r tr roots n vs v n rursvy r. Aorn to Corory 6, G s y tru s rturn. Oprton (MrA) Wn MrA s n AG, t oown r pror:. I tr xsts no n tt s not n r, tn rturns s.. Otrws, s no AG, t rturns tru. I s AGs, s MrA n AG.. I ny AG rturns s (wt unr nos), tn rturns s. Otrws, rturns tru. Fny, t top v oprton TstAyty ons t prvousy n oprtons to vry t yty o G. Oprton (TstAyty) Wn TstAyty s ntt n yprtr AG unon G, t oown r pror: 0
. A AG s rtrry osn s t root o t yprtr.. s PrPross n ts.. s MrNo n ts rpty unt s s rturn (no no s r n t st ). 4. s MrA n ts. I tru s rturn, tn TstAyty rturns y (G s y). Otrws, rturn y. T oown tor stss t orrtnss o t ort. Tor T oprton TstAyty orrty trns t yty o yprtr AG unon. Proo: Aorn to Corory 6, t s sunt to r non--spno roots/vs n - spno roots/vs rursvy n trnty. PrPross os t rst roun o rursv rn o non--spno roots/vs, n rpt MrNo prors t susqunt rursv n trnt rn. E MrNo nts -spno roots/vs y CotFyIno (tr p =0or = 0) n tn rs t s w s nw non-spno roots/vs y strutmr. By Corory 6, MrNo w not rturn s unt roots n vs r r. MrA tsts t AG unon s pty n w trn t yty orrty. W ustrt t prorn o TstAyty wt two xps, y AG unon n n y on. T rst s t AG unon pt n Fur 4. Suppos s st s t root. Atr PrPross t unon oos s Fur 6. Wn s MrNo n ts, t s n ts CotFyIno() w s tn propt to. rturns trp (,, 0) to. Susqunty, nrts t n trp (,, ) n trnts CotFyIno(). Sn ntr p nor s zro, strutmr() s not. tn s n ts CotFyIno() w vntuy trnts sry s CotFyIno(). s MrNo n. s n ts CotFyIno() w s tn propt to. rturns trp (,, 0) to. Susqunty, nrts t n trp (,, ) n trnts CotFyIno(). NostrutMr s. tn s n ts CotFyIno() w vntuy trnts sry s CotFyIno(). s MrNo n w rturns s sn s no AG. Ts uss to rturn s to w so rturns s n trnts MrNo. s MrA n ts n rturns s ty. TstAyty tn trnts wt y rturn.
n o Fur 7: An y AG unon. As notr xp, onsr t AG unon n Fur 7. It s nt to tt n Fur 4 xpt tt t r ro to s now rvrs. o () n n o () () n o () o n n o () Fur 8: Prorn o TstAyty n t AG unon o Fur 7. Suppos s st s t root. Fur 8 () sows t unon tr PrPross. Not tt nos n r sot n ut nnot r urn PrPross. PrPross s so n sn n sot -spno n on AG y st prtpt n rt y n otr AGs. It wou nonsstnt to r t n on AG n to p t unr n notr. Wn MrNo s rst n wtn TstAyty, t s CotFyIno() n ts n tn CotFyIno() s n t prvous xp.
s MrNo n. s n ts CotFyIno() w s tn propt to. rturns trp (, 0, 0) to. Susqunty, nrts t n trp (, 0, ) n trnts CotFyIno(). Sn p =0, s strutmr() n ts. It rs n, n tn s strutmr() n w rs s w. Fur 8 () sows t rsutnt unon. tn s n ts CotFyIno() w vntuy trnts sry s CotFyIno(). Fur 8 () sows t rsutnt unon. s MrNo n w rturns s. Sn strutmr ws n, t rturns tru to w rturns tru n trnts t rst o MrNo n TstAyty. Wn MrNo s n t son t wtn TstAyty, s n ts CotFyIno() w s tn propt to. rturns trp (, 0, 0) to. Susqunty, nrts t n trp (, 0, ) n trnts CotFyIno(). Sn p =0, s strutmr() n ts. Ts uss t rn o,, n n n. Fur 8 () sows t rsutnt unon. tn s n ts CotFyIno(). It vntuy trnts sry s CotFyIno() wt r n ot n. Fur 8 () sows t rsutnt unon. s MrNo n w tn s MrNo n. s no AG n rturns s, w uss to rturn s to. Sn strutmr ws n, t rturns tru n trnts t son o MrNo n TstAyty. Wn MrNo s n t tr t wtn TstAyty, t propts MrNo to n tn to. Evntuy, s s rturn. Wn MrA s n, t propts t oprton to t rst o t unon. Evntuy, tru s rturn. T oprton TstAyty trnts wt y rturn. 7 Copxty Anyss W not t xu nur o nos n AG y, t xu nur o nt nos o no n AG y t, t xu nur o nos n -spst y, t xu nur o AGs tt y ontn -spno y s, n t tot nur o AGs n t yprtr AG unon y n. To rursvy r non--spnos n AG, O(t ) nos n to. Hn PrPross s O(n t) nos. E CotFyIno (rtv to sn -spno) tsts -spno n O(s) AGs. To r -spno, O(s) AGs pror t rn, o w so s
O(t ) non--spnos. Hn t opxty o CotFyIno n strutmr or r -spno s O(s t). T opxty o MrNo ro TstAyty s tn O(n st). Sn t st on -spno w r or o MrNo, Mrno w O(n ) ts. Hn t opxty o MrNo s s O(n st). MrA s O(n) nos. Tror, t worst s opxty o TstAyty s O(n st). 8 susson In ts ppr, w prsnt n nt strut ort, or vrton o yty o t ovr strutur o MSBN. An portnt tur o t ort s tt t os not rqur sunt/nt n t syst to rv ts ntrn strutur. Fro t nton o CotFyIno, ry nt ony provs norton rrn wtr sr no s ny prnt or n t AG tt t nt s rspons or. Tror, t ort supports t onstruton o MSBNs onstrut ro utp oputton nts ut y utp vnors w provn utot vrton o orrtnss o t ovr strutur. Our strut ort s s on Corory 6 w n turn s s on Proposton 5. Proposton 5 xtns t o topoo sortn n tt t ttr s quvnt to rn ony root nos. Evn n ntrz tst, Proposton 5 ows or nt tst tn topoo sortn. Ts s us Proposton 5 ts on vr tw s ny r nos t rursv rn, vn ss nos to n susqunt rn prosss. Atou topoo sortn n so xtn nto strut ort, t vnt o TstAyty s or pronnt n ooprtv tst. Sn MrNo rs or nos n roun tn topoo sortn os, ny ss s o MrNo w (u or nt ooprton), w trnsts nto ruton n ounton ovr. As nton n Ston, strut vrton tts yn orton o MSBN. In nr, TstAyty sou pror wnvr t r sunts o MSBN n. Howvr, tr r sp ss wr t xuton o u s TstAyty s unnssry. For xp, nw sunt ony ntrs wt on xstn sunt n t -spst twn t ontns root nos ony, tn t yty o t nw AG unon n onr oy. On usu rton or utur rsr s to nty su sp ss n to vop sp vrton oprtons orny. Anownts Ts wor s support y t Rsr Grnt OGP05545 ro t Ntur Sns n Ennrn Rsr Coun (NSERC) o Cn. Hpu onts on n rr rt ro T. Cu n S. Huntr r now. 4
Rrns [] E. Crn. Bysn ntwors wtout trs. AI Mzn, (4):50 6, 99. [] V.R. Lssr n.. Cor. Funtony urt, ooprtv strut systs. IEEE Trns. on Systs, Mn n Cyrnts, SMC-():8 96, 98. [] R.E. Npotn. Prost Rsonn n Exprt Systs. Jon Wy n Sons, 990. [4] J. Pr. Prost Rsonn n Intnt Systs: Ntwors o Pus Inrn. Morn Kunn, 988. [5] S. Srnvs. A prost ppro to rr o-s noss. In Pro. 0t Con. Unrtnty n Art Intn, ps 58 545, Stt, Wsnton, 994. [6].F. Stus n N.W. Wr. t Struturs wt Astrt t Typs n Mou-. Broos/Co, 987. [7] Y. Xn. Optzton o ntr-sunt uptn n utpy ston Bysn ntwors. In Pro. t Con. on Unrtnty n Art Intn, ps 565 57, Montr, 995. [8] Y. Xn. A prost rwor or ooprtv ut-nt strut ntrprtton n optzton o ounton. Art Intn, 87(-):95 4, 996. [9] Y. Xn. Snts o utpy ston Bysn ntwors or ooprtv ut-nt strut ntrprtton. In G. MC, tor, Avns n Art Intn, ps 6. Sprnr, 996. [0] Y. Xn, B. Pnt, A. Esn, M. P. Bos, n. Poo. Mutpy ston Bysn ntwors or nurousur noss. Art Intn n Mn, 5:9 4, 99. [] Y. Xn,. Poo, n M. P. Bos. Exporn oty n Bysn ntwors or r xprt systs. In Pro. 8t Con. on Unrtnty n Art Intn, ps 44 5, Stnor, 99. [] Y. Xn,. Poo, n M. P. Bos. Mutpy ston Bysn ntwors n unton orsts or r now s systs. Coputton Intn, 9():7 0, 99. 5